05_Numerical_Differentiation.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Lecture 4: Numerical Differentiation\n",
    "===\n",
    "\n",
    "Finite difference approximations\n",
    "---\n",
    "\n",
    "Forward differences\n",
    "$$\n",
    "f'_{FD}(x, h) = \\frac{f(x + h) - f(x)}{h}\n",
    "$$\n",
    "\n",
    "Backward differences\n",
    "$$\n",
    "f'_{BD}(x, h) = \\frac{f(x) - f(x-h)}{h}\n",
    "$$\n",
    "\n",
    "Central differences\n",
    "$$\n",
    "f'_{CD}(x, h) = \\frac{f(x+h) - f(x-h)}{2h}\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "true"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "using PyPlot\n",
    "PyPlot.svg(true)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "fprime_CD (generic function with 1 method)"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fprime_FD(f, x, h) = (f.(x .+ h) - f.(x))/h\n",
    "fprime_CD(f, x, h) = (f.(x .+ h) - f.(x .- h))/(2*h)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "FD 3.425517647315246\n",
      "CD 3.4255187575382706\n",
      "exact 3.42551882081476\n"
     ]
    }
   ],
   "source": [
    "f(x) = tan.(x)\n",
    "\n",
    "h = 1e-10\n",
    "\n",
    "fd = fprime_FD(f, 1, h)\n",
    "cd = fprime_CD(f, 1, h)\n",
    "exact = 1 + tan(1)^2\n",
    "\n",
    "println(\"FD \", fd)\n",
    "println(\"CD \", cd)\n",
    "println(\"exact \", exact)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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   ],
   "source": [
    "# error analysis\n",
    "hs = [1e-1, 1e-2, 1e-3, 1e-4, 1e-5, 1e-6, 1e-7, 1e-8, 1e-9, 1e-10, 1e-11, 1e-12]\n",
    "\n",
    "fds = [fprime_FD(f, 1, h) for h in hs]\n",
    "cds = [fprime_CD(f, 1, h) for h in hs]\n",
    "\n",
    "errors_fd = abs.((fds .- exact)./exact)\n",
    "errors_cd = abs.((cds .- exact)./exact)\n",
    "\n",
    "figure()\n",
    "loglog(hs, errors_fd, \"o-\", label=\"Forward difference\")\n",
    "loglog(hs, errors_cd, \"o-\", label=\"Central difference\")\n",
    "\n",
    "xlabel(\"h\")\n",
    "ylabel(\"relative error\")\n",
    "legend()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Automatic Differentiation\n",
    "---\n",
    "\n",
    "We implement Forward Mode automatic differentiation using dual numbers.\n",
    "\n",
    "Note: For your own programs you can use the ForwardDiff.jl and ReverseDiff.jl packages,\n",
    "which support most Julia code."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "# defining dual numbers\n",
    "struct DualNumber\n",
    "    x::Float64\n",
    "    xdot::Float64\n",
    "end\n",
    "\n",
    "# overloading plus\n",
    "function Base.:+(a::DualNumber, b::DualNumber)\n",
    "    return DualNumber(a.x + b.x, a.xdot + b.xdot)\n",
    "end\n",
    "\n",
    "# overloading minus\n",
    "function Base.:-(a::DualNumber, b::DualNumber)\n",
    "    return DualNumber(a.x - b.x, a.xdot - b.xdot)\n",
    "end\n",
    "\n",
    "# overloading times\n",
    "function Base.:*(a::DualNumber, b::DualNumber)\n",
    "    return DualNumber(a.x*b.x, a.x*b.xdot + a.xdot*b.x)\n",
    "end\n",
    "\n",
    "function Base.:*(a::Real, b::DualNumber)\n",
    "    return DualNumber(a*b.x, a*b.xdot)\n",
    "end\n",
    "\n",
    "# overloading divide\n",
    "function Base.:/(a::DualNumber, b::DualNumber)\n",
    "    return DualNumber(a.x/b.x, (a.xdot*b.x - a.x*b.xdot)/(b.x)^2)\n",
    "end\n",
    "\n",
    "# overloading a few basic functions\n",
    "function Base.exp(a::DualNumber)\n",
    "    return DualNumber(exp(a.x), a.xdot*exp(a.x))\n",
    "end\n",
    "\n",
    "function Base.sin(a::DualNumber)\n",
    "    return DualNumber(sin(a.x), a.xdot*cos(a.x))\n",
    "end\n",
    "\n",
    "function Base.cos(a::DualNumber)\n",
    "    return DualNumber(cos(a.x), -a.xdot*sin(a.x))\n",
    "end\n",
    "\n",
    "# comparison with a real number\n",
    "function Base.:<(a::Real, b::DualNumber)\n",
    "    return a < b.x\n",
    "end\n",
    "\n",
    "function Base.:<(a::DualNumber, b::Real)\n",
    "    return a.x < b\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "DualNumber(1472.3488987227095, 10198.083477451979)"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a = DualNumber(1.0, 1.0)\n",
    "\n",
    "# a complicated function\n",
    "f(x) = exp(exp(x))*sin(x) + exp(cos(x) + sin(x))*sin(cos(sin(x)))\n",
    "\n",
    "f(DualNumber(2.0, 1.0))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
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   "source": [
    "# Computing the derivative of a simple function: tan(x) = sin(x)/cos(x)\n",
    "f(x) = sin(x)/cos(x)\n",
    "\n",
    "xs = LinRange(-1, 1, 100)\n",
    "xduals = [DualNumber(x, 1.0) for x in xs]\n",
    "dual_results = f.(xduals)\n",
    "\n",
    "fs = [d.x for d in dual_results]\n",
    "fprimes = [d.xdot for d in dual_results]\n",
    "\n",
    "figure()\n",
    "plot(xs, fs, label=\"f(x)\")\n",
    "plot(xs, fprimes, label=\"f'(x)\")\n",
    "\n",
    "xlabel(\"x\")\n",
    "legend()"
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  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Comparing accuracy between AD, manual evaluation, and finite differences"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
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    "# comparing to the exact solution\n",
    "fprime(x) = 1.0 .+ tan.(x).^2\n",
    "\n",
    "fprimes_exact = fprime(xs)\n",
    "\n",
    "figure()\n",
    "plot(xs, abs.(fprimes - fprimes_exact))\n",
    "\n",
    "xlabel(\"x\")\n",
    "ylabel(L\"|f'(x) - f'_{AD}(x)|\")"
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  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "complicated (generic function with 1 method)"
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     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Autodiff'ing program code\n",
    "\n",
    "function complicated(x)\n",
    "    if x > 1\n",
    "        x = x*x - x\n",
    "    else\n",
    "        x = x*x - x - x\n",
    "    end\n",
    "    \n",
    "    for i=1:3\n",
    "        if x < 1\n",
    "            x = x*x + exp(x)*sin(x) \n",
    "        else\n",
    "            x = 0.9x\n",
    "        end\n",
    "    end\n",
    "    \n",
    "    return x\n",
    "end"
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  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "# what does this function do?\n",
    "\n",
    "xs = LinRange(-1, 1, 100)\n",
    "ys = complicated.(xs);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
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      "text/plain": [
       "Figure(PyObject <Figure size 640x480 with 1 Axes>)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "PyObject Text(26.000000000000007, 0.5, '$f(x)$')"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "figure()\n",
    "plot(xs, ys)\n",
    "\n",
    "xlabel(L\"x\")\n",
    "ylabel(L\"f(x)\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "# autodiff can handle it!\n",
    "\n",
    "xduals = [DualNumber(x, 1) for x in xs]\n",
    "dual_results = complicated.(xduals)\n",
    "\n",
    "fprimes = [d.xdot for d in dual_results];"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
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